| L(s) = 1 | − 5-s + 7-s − 6·13-s + 7·17-s − 8·19-s + 2·23-s − 4·25-s − 4·29-s − 2·31-s − 35-s + 9·37-s − 11·41-s − 7·43-s + 11·47-s + 49-s − 10·53-s − 9·59-s − 8·61-s + 6·65-s + 12·71-s − 14·73-s + 79-s + 9·83-s − 7·85-s − 14·89-s − 6·91-s + 8·95-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.377·7-s − 1.66·13-s + 1.69·17-s − 1.83·19-s + 0.417·23-s − 4/5·25-s − 0.742·29-s − 0.359·31-s − 0.169·35-s + 1.47·37-s − 1.71·41-s − 1.06·43-s + 1.60·47-s + 1/7·49-s − 1.37·53-s − 1.17·59-s − 1.02·61-s + 0.744·65-s + 1.42·71-s − 1.63·73-s + 0.112·79-s + 0.987·83-s − 0.759·85-s − 1.48·89-s − 0.628·91-s + 0.820·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 9 T + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 - 11 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.132846287495617390673357485015, −8.009679383935202880511542786667, −7.69066101718117845416496839660, −6.73832228944977989771565810879, −5.69063766258212549783384841332, −4.85394111651867064243188369421, −4.01865922045451307330014065016, −2.89937084612398639042749934040, −1.76165475797795108247757840445, 0,
1.76165475797795108247757840445, 2.89937084612398639042749934040, 4.01865922045451307330014065016, 4.85394111651867064243188369421, 5.69063766258212549783384841332, 6.73832228944977989771565810879, 7.69066101718117845416496839660, 8.009679383935202880511542786667, 9.132846287495617390673357485015
